So why bother with calculus? Well, there’s all sorts of really good reasons for knowing about calculus and using calculus. And one of

the best things about calculus is derivatives, and here’s a good example of

how to use a derivative in an everyday engineering application. We’ll take a look

at a dial right here. So a dial turns like that and it turns around its center. And

we can describe the motion of turning from one number to another as something called a “sigmoid.” A sigmoid, if we graph it, looks something like this. This is a graph of time versus angle (angle being

“theta”, in this case) If I take the motion of a

typical dial moving from one angle to another it would look something like

this. It’s what we call a sigmoid. Now, a sigmoid starts at an angle —

so this is a start right here — and it will end at an angle right there. So, start and end. And the thing is about a sigmoid that’s kind of interesting is its slowest

at the end and slowest the beginning. But right in the middle — right here — it’s really fast. So we’ve spoken about the concept of the movement of the dial as a sigmoid. Now, let’s actually write that sigmoid

function out in a program called Maple. And so what you can see is the actual

equation right here. And I will create a plot of it like this so you can see that the

plot in Maple looks similar to the hand-drawn plot and I had before. We can go even

one better than that. So if I right-click on that equation I can say “differentiate with respect to time” (or “with respect to ‘t'”). And so I’ve got the time derivative of that sigmoid function. I can also plot that. I right click on the

equation and I do a plot like this… So I’m going to colour this plot green… and I’m going to make that line a little

bit wider… like that … and do the same thing that sigmoid right here. I’m going to make that line

wider… like that. Now, what I’m going to do is superimpose one on top of

the other, so you can see both the sigmoid and it’s time derivative. So the

sigmoid is a dark red and the time derivative is in green.

And what you can see is the maximum speed or the maximum derivative is always found at the center

of that sigmoid. So now that we have this concept down let’s take a look at

what happens if we actually implement this in real electronics hardware. So now

we’re ready to implement real calculus on real hardware. What we have here is

a microprocessor, we have a dial (or “potentiometer”) right here and then some

LEDs right beside the dial. And what will happen is that there’s code inside this

microprocessor that will read the angle — for which I’m turning this dial right here —

and the faster I turn the dial or the greater the derivative of the motion of

the dial, the more LEDs will turn on. So if I go REALLY slow… one LED turned on. A

little bit faster I get two LEDs that turned on. If I go really fast, three LEDs. Reset it.

Really fast, I get four! And so the way this works, basically, is I’ve

got code written up that configures four LEDs. I have some code, as well, that measures time. And I’ve got specific code to

read the value of that dial. And then I’ve got some code in here that calculates the

difference in time from one measurement to another measurement. And I’ve got code

here that calculates the difference in angle effectively from one measurement

to another measurement. And I take those two differences — the change in time and the

change in angle — and I can create a derivative from it. From there I just

have some code that turns on a bunch of LEDs. And it’s _that_ simple. So basically whatI do is I reset it right here. I turn the dial really slowly, like that, and one or two

LEDs turns on… or really fast and four LEDs turn on. This is the actual implementation

of actual calculus, an actual derivative, on actual hardware.

Very good and easy explanation!. Some times the hard part is to visualize the equations into a real solution.